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A Sasaki manifold is K-semistable whenever its CR Yamabe invariant equals the minimum of the Einstein-Hilbert functional on the Reeb cone.

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If the CR Yamabe invariant of a Sasaki manifold attains the minimum determined by its Reeb cone, the manifold is K-semistable, linking CR analysis to algebraic stability.

T0 review reviewed 2026-08-05 challenge →

load-bearing objection A plausible and genuinely new link between CR Yamabe invariant and Sasaki K-stability, but the key analytic lemma is only sketched. the 3 major comments →

arxiv 2509.00743 v1 pith:FDEG342A submitted 2025-08-31 math.DG math.AG

The CR Yamabe invariant and constant scalar curvature Sasaki metrics

Abdellah Lahdili , Eveline Legendre , Carlo Scarpa This is my paper
classification math.DG math.AG MSC 53C2553D10
keywords CR Yamabe invariantSasaki geometryconstant transversal scalar curvatureK-semistabilitySasaki-Futaki invariantEinstein-Hilbert functionaltest configurationscscK metrics
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper tries to establish a bridge between the CR Yamabe invariant of a compact Sasaki manifold and the existence and stability of constant scalar curvature Sasaki (cscS) metrics. Its central claim is that if the CR Yamabe invariant attains the minimum value of the Einstein-Hilbert functional on the Sasaki-Reeb cone, then the Sasaki manifold is K-semistable, meaning every test configuration has nonnegative Sasaki-Futaki invariant. The authors show that equality is also detected by approximate cscS metrics when the average scalar curvature is nonpositive, and they give a numerical criterion for K-semistability of polarized complex manifolds. The approach matters because it detects cscS structures without first prescribing their Reeb vector fields, and it provides a Yamabe-style route to the algebraic stability side of the cscS/cscK existence problem.

Core claim

On the circle bundle over a polarized manifold, the paper defines the CR Yamabe energy Y_CR(η)=inf_f EH(f^{-1}η) and shows its critical points are exactly cscS structures whose Reeb field minimizes the Einstein-Hilbert functional EH on the Sasaki-Reeb cone. The invariant Y_sup^T(X,L)=sup_η Y_CR^T(η) is then compared with EH_min. The main theorem: if Y_sup^T=EH_min, the Sasaki manifold (N,I,χ_min) is K-semistable—every test configuration has nonnegative Sasaki-Futaki invariant. The proof refines Y_sup^T≤EH_min to Y_sup^T≤EH_s^χ(X,L) for smooth ample dominant test configurations, and the s-derivative of EH_s^χ at s=0 is the Sasaki-Futaki invariant, so equality forces nonnegativity. When EH_min

What carries the argument

The load-bearing object is the CR Yamabe energy Y_CR(η)=inf_f EH(f^{-1}η), where EH(α) is the CR Einstein-Hilbert functional (total Tanaka-Webster scalar curvature divided by a power of volume) on the conformal class of a Sasaki contact form η. Varying η over Kähler forms in c1(L) gives the invariant Y_sup^T(X,L). The key inequality chain is Y_sup^T≤EH_s^χ(X,L) for test configurations, obtained by extending EH along weak geodesic ribbons via an action functional; differentiating EH_s^χ at s=0 recovers the Sasaki-Futaki invariant of the test configuration. Equality of the two ends of the chain converts a curvature-bound inequality into an algebro-geometric stability statement.

Load-bearing premise

The argument's load-bearing premise is that the action functional along the contact-form families induced by test configurations is convex and satisfies a slope inequality; the paper only sketches these proofs and defers them to earlier work, so the equality-to-K-semistability implication would fail if that deferred analysis is wrong.

What would settle it

On the toric circle bundles over P^1×P^1 with polarization pO(2)+qO(2), q>5p, discussed in Section 2.4, compute the CR Yamabe invariant from the L^{n+1} formula of Theorem 1.6 over the explicit toric Sasaki metrics. If Y_sup equals EH(ξ0) while some explicit test configuration has negative Sasaki-Futaki invariant, Theorem 1.4 is contradicted; if Y_sup is strictly below EH(ξ0), the equality criterion has real content and can be compared with the known three cscS structures.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • If Y_sup^T(X,L)=EH_min, the Sasaki manifold (N,I,χ_min) is K-semistable: all Sasaki-Futaki invariants of T-equivariant test configurations are nonnegative.
  • A single contact form η with Y_CR^T(η)=EH_min produces an actual cscS metric, η(χ_min)^{-1}η, with Reeb field a minimizer of EH; when EH_min≤0, the converse holds.
  • When EH_min≤0, |Y_sup^T| equals the infimum over conformal classes of the L^{n+1} norm of Tanaka-Webster scalar curvature, giving a numerical handle on the invariant.
  • Under EH(ξ0)=EH_min and c1(X)·c1(L)^{n-1}≤0, the existence of L^p-approximate cscK metrics implies K-semistability of the polarized manifold (X,L).
  • A cscS structure satisfying the eigenvalue bound λ_1^T>c_η/2(n+1) is isolated in the T-invariant space of cscS structures.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The equality Y_sup^T=EH_min is probably closer to an existence criterion than to a semistability criterion: in the nonpositive case the paper proves equality from approximate solutions, so a full converse would make CR Yamabe equality equivalent to the existence of cscS metrics, i.e. a weak existence-stability correspondence for this curvature condition.
  • Because only minimizers of EH are detected, cscS structures whose Reeb fields are non-minimizing (like the three toric examples on P^1×P^1) will not be visible to the CR Yamabe invariant; a moduli or isolation theory would need a different functional.
  • The L^{n+1}-norm characterization invites explicit computations of Y_sup on toric or join Sasaki manifolds with negative average curvature; such computations could turn the K-semistability criterion into a practical numerical test.
  • The stated regularity assumption (existence of a regular Reeb vector field in the cone) is likely unnecessary for the K-stability part: the same action-functional argument should extend to quasi-regular orbifold quotients, as the paper itself hints.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper studies the CR Yamabe energy on the Boothby-Wang circle bundle N associated to a polarized manifold (X,L), and connects it with constant transversal scalar curvature Sasaki (cscS) structures and Sasaki K-stability. The main invariant is Y^T_sup(X,L), the supremum of the equivariant CR Yamabe energy over Sasaki forms with a fixed regular Reeb field, while EH_min is the minimum of the Einstein-Hilbert functional over the Sasaki-Reeb cone. The central results are: (i) if Y^T_sup(X,L)=EH_min, then the Sasaki manifold with the minimizing Reeb field is K-semistable (Theorem 1.4); (ii) under non-positive average scalar curvature, existence of L^p-approximate cscS structures implies the equality Y^T_sup=EH_min (Corollary 1.8); (iii) a partial converse is discussed but explicitly not completed. The proof of Theorem 1.4 proceeds through an inequality Y^T_sup(X,L) ≤ EH^χ_s(X,L) for test configurations, proved using an action functional along weak geodesic ribbons. The paper also contains regularity results for the CR Yamabe energy and an isolation theorem for cscS structures.

Significance. If fully established, the paper gives a new and potentially important bridge between the CR Yamabe invariant and Sasaki K-stability, with a concrete numerical criterion for K-semistability of polarized manifolds. The approach is original: it uses the global conformal invariant rather than prescribing a Reeb vector field. Several computations are explicit and non-tautological, e.g. the test-configuration slopes in Propositions 5.12 and 5.13, and the example on P^1×P^1 is a nice illustration. However, the central implication rests on analytic convexity and slope results (Theorem 5.6 and Proposition 5.7) that are only sketched and deferred to the authors' related work and to Inoue. Until those arguments are supplied, the main theorem is conditional.

major comments (3)
  1. [§5.1–5.2, Theorem 5.6 and Proposition 5.7] These two results are the load-bearing analytic inputs for Corollary 5.8 and hence Theorem 1.4. The text only sketches Theorem 5.6 as a consequence of Proposition 5.11. But the formula for dd^c A^Ψ_χ contains the term ns∫ dd^c Ψ∧(ω+ddcΦ)^n / f^{n+1}; no argument is given that this term is nonnegative, nor is an integration-by-parts or sign estimate provided. Moreover, the passage from the regularized functional A^Ψ to A by approximating metrics is asserted to be 'exactly the same as in [Ino21, LLS23]' but is not carried out. Since [LLS23] corresponds to the regular Reeb case, the new non-regular statement is not actually proved here.
  2. [§5.1, proof of Theorem 1.4] The proof of K-semistability only treats smooth, ample, dominant T-equivariant test configurations with reduced central fibre. The definition of Sasaki K-semistability in [CS18] concerns all test configurations; the manuscript does not justify that the restricted class is sufficient. If this reduction is standard, a precise reference or a short argument is needed; otherwise the conclusion of Theorem 1.4 exceeds what has been shown.
  3. [§4.1, approximate cscS converse] The abstract advertises 'a partial converse' to Corollary 1.8, but Section 4.1 explicitly states 'We are not yet able to obtain such bounds' after Lemma 4.5. Lemmas 4.4 and 4.5 give only weak estimates and do not imply the existence of L^p-approximate cscS structures. The reader should be told precisely which statement is conjectural and which is proved; as written, the advertised partial converse is not a theorem.
minor comments (5)
  1. [Introduction vs. §5.1] The inequality Y^T_sup(X,L) ≤ EH^χ_s(X,L) is stated as Theorem 1.5 in the Introduction but appears as Corollary 5.8 in Section 5.1. Please renumber consistently.
  2. [§2.1, proof of Proposition 2.3] The case n=1 requires separate wording: q=4 and the Sobolev embedding W^{1,2}⊂L^q is compact on the 2-dimensional quotient, but this is not made explicit. The current sentence comparing q with the critical Sobolev exponent assumes n>1.
  3. [§5.2, after Proposition 5.11] The phrase 'we freely use results proved for A^Ψ_χ, such as Proposition 5.11, for Aχ without further comment' is too quick; the approximation argument should at least state what convergence properties are preserved (lower semicontinuity, convexity, slopes).
  4. [References] The reference [RT1105] appears to have a formatting error in the year/volume fields. Please correct.
  5. [Throughout] There are several typos and grammatical slips, e.g. 'of of' near the beginning of Section 2, and 'The converse holds, when EH_min ≤0' with an awkward comma. These do not affect the mathematics.

Circularity Check

0 steps flagged

No circular derivation found; the central implication rests on nontrivial analytic lemmas partly deferred to prior work, not on a definitional or fitted equivalence.

full rationale

The paper's central claim, Theorem 1.4, is an implication: if Y^T_sup(X,L) = EH_min, then the Sasaki manifold is K-semistable. The inequality Y^T_CR(η) ≤ EH(χ) for all χ ∈ t+ is indeed built into the definition of the CR Yamabe energy as an infimum over a conformal class, and Y^T_sup ≤ EH_min is immediate. But the equality condition is a genuine assumption, not a definitional restatement: it asks that the sup over all transverse CR structures coincide with the infimum of the Einstein-Hilbert functional over Reeb fields. That is not a fitted parameter or a renamed conclusion. The proof of Theorem 1.4 proceeds through Corollary 5.8, which establishes Y^T_CR(X,L) ≤ EH^χ_s(X,L) for suitable test configurations. The proof uses convexity of the action functional along weak geodesic ribbons (Theorem 5.6), the slope inequality (Proposition 5.7), and the constancy of volume along ribbons (Lemma 5.9). These are analytic results, not identities manufactured from the target statement. Lemma 5.4 identifies the first-order coefficient of EH^χ_s with the global Sasaki-Futaki invariant of [ACL21]; this is a computation about a previously defined algebraic invariant, not a tautology. The paper does rely on the authors' own prior work [LLS23] and on [Ino21] for technical parts of the argument: Theorem 5.6 is stated as a generalization of [LLS23, Theorem 1.4], and the passage from the modified action functional A^Ψ to A is said to follow by an approximation argument 'exactly the same as in [Ino21, LLS23]'. Some of these supporting results are only sketched here, and the text explicitly refers to [LLS23] for details. This is a real support gap and a correctness risk, especially for non-regular Reeb fields, but it is not circularity: the cited results are not shown to be equivalent to the paper's conclusion, and no step reduces the desired K-semistability to its own assumption by definition. Similarly, the implicit restriction to smooth, ample, dominant test configurations with reduced central fibre in concluding K-semistability is a logical gap, not a circular reduction. There are no fitted parameters presented as predictions, no renaming of a known empirical pattern as a new structure, and no uniqueness theorem imported from the authors' prior work to forbid alternatives. The self-citations are load-bearing for technical lemmas but do not make the main claim equivalent to its inputs. Accordingly, the circularity score is low, reflecting only the heavy reliance on unverifi

Axiom & Free-Parameter Ledger

0 free parameters · 6 axioms · 0 invented entities

No free parameters. The paper's central claim rests on standard results in CR and Kähler geometry, on the authors' earlier technical machinery [LLS23], and on the standing assumption that the Sasaki-Reeb cone contains a regular vector field. The K-semistability conclusion also implicitly assumes a reduction to smooth test configurations that is not stated.

axioms (6)
  • domain assumption The equivariant CR Yamabe minimizer exists in every T-invariant conformal class and is smooth when the class contains a regular Sasaki form.
    Used in Proposition 2.3 and Theorem 2.2; requires regularity of the Reeb field to reduce to a subcritical PDE on the Kähler quotient (Section 2.1).
  • domain assumption EH attains a minimum on the Sasaki-Reeb cone t+.
    Used to define EHmin and chi_min; cited from [BHL18] in Section 2.3.
  • standard math The action functional A_chi(s,t) is convex and lower semicontinuous along weak geodesic ribbons, and its slope at t=0 is bounded below by total scalar curvature.
    Theorems 5.6 and Proposition 5.7; proofs are sketched and adapted from [LLS23] and [Ino21] (Section 5.2).
  • domain assumption K-semistability of the Sasaki manifold can be tested on smooth, ample, dominant T-equivariant test configurations with reduced central fibre.
    The proof of Theorem 1.4 checks only this class; the paper does not state the reduction from arbitrary test configurations (page 31).
  • domain assumption The derivative (d/ds)|_{s=0} EH^chi_s equals 2n V^{n/(n+1)} SF, with SF the global Sasaki-Futaki invariant of [ACL21].
    Lemma 5.4; the definition of SF is taken from [ACL21] and the paper proves the derivative formula via Theorem 5.3.
  • domain assumption The Sasaki-Reeb cone t+ contains at least one regular vector field (xi0 from the Boothby-Wang fibration).
    Assumed in the introduction and Section 5.2: 'which shows why we assume that there exists at least one regular Reeb vector field'; used to identify P(N,I,xi)^T with P(X,L)^T.

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Pith. "Pith review of The CR Yamabe invariant and constant scalar curvature Sasaki metrics." pith.science (2026). https://pith.science/paper/FDEG342A

@misc{pith2026250900743,
  author       = {Pith},
  title        = {Pith review of: The CR Yamabe invariant and constant scalar curvature Sasaki metrics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FDEG342A}},
  note         = {Machine review of arXiv:2509.00743}
}
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read the original abstract

We propose a new approach to the existence of constant transversal scalar curvature Sasaki structures drawing on ideas and tools from the CR Yamabe problem, establishing a link between the CR Yamabe invariant, the existence of Sasaki structures of constant transversal scalar curvature, and the K-stability of Sasaki manifolds. Assuming that the Sasaki-Reeb cone contains a regular vector field, we show that if the CR Yamabe invariant of a compact Sasaki manifold attains a specific value determined by the geometry of the Reeb cone, then the Sasaki manifold is K-semistable. Under the additional assumption of non-positive average scalar curvature, the CR Yamabe invariant attains this topological value if the manifold admits approximately constant scalar curvature Sasaki structures, and we also show a partial converse. As an application, we provide a new numerical criterion for the K-semistability of polarised compact complex manifolds.

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Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. The toric CR Yamabe problem

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    On any compact toric contact manifold of Reeb type, T-invariant CR structures realize both positive and negative CR Yamabe invariants, via a polytope PDE reduction.

  2. Yamabe-type problems on compact Hermitian manifolds

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This paper was first reviewed by deepseek-v4-flash on August 5, 2026.